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Recombining Building Blocks (again) in simple building block functions and in natural genomes or How two scales of optimisation can be better than one.

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Presentation on theme: "Recombining Building Blocks (again) in simple building block functions and in natural genomes or How two scales of optimisation can be better than one."— Presentation transcript:

1 Recombining Building Blocks (again) in simple building block functions and in natural genomes or How two scales of optimisation can be better than one or Another real royal road Richard A. Watson Natural Systems group ECS, Southampton University, UK

2 blocks - 2 Richard Watson Overview Work in progress, no clever analysis, just some concepts Relationship between sex, problem decomposition, coev. Discussion of concatenated BB functions Mutation vs crossover in fully-deceptive blocks When is a block not a block? (when its an atom with a big alphabet) Two scales of optimisation in partially deceptive subfunctions What does that mean for a biological model? Generalisations Natural properties of genomes

3 blocks - 3 Richard Watson Objectives (wish list) Understand sufficient conditions (if any) for evolving population to be required to use building block combination A ‘Royal Road’, based on building blocks, allowing use of linkage information – and, that isn’t too contrived Specifically, is there a kind of problem space where GA >> hill climber (same assumptions of epistasis and linkage), that uses population for combination of building blocks/search combinations of blocks/move in block-sequence space (not just preserving ‘common good’) And, keep it simple/intuitive!  biologically plausible e.g. not HIFF (Sudholt), not gap function, not intersecting ridges

4 blocks - 4 Richard Watson Some thoughts about contrivance In general Some (diverse) points in genotype space are easy to reach Recombination of genotypes from these areas results in a jump to a new genotype that is high in fitness, this peak is otherwise unreachable Why is the peak just where it is? The building block intuition is that A-good, B-good→A+B-better A-easy, B-easy→A+B-easy??

5 blocks - 5 Richard Watson Intersecting ridges klogk block 1 block  --- block 1 block  block 1 block block 1 block 2 T klogk

6 blocks - 6 Richard Watson An old favourite: Concatenated trap functions i.e. Separable sub-functions With sub-optimal peak in each sub-function In much prior work with these it is supposed that: Selection on bits not sufficient and not necessary Selection on blocks is necessary and sufficient precluding all utility of selection on bits, as in fully- deceptive trap functions, maximises the advantage of selecting on blocks (?) e.g. unitation 012… k fitness k-1

7 blocks - 7 Richard Watson However If the selective gradient within a block is not useful, then all means to find a good block are going to be exponential in k. Hence popsize needs to be exp. in k. Hence assumption that blocks size, k, must be small (constant, not fraction of L) Afterall, if algorithmic advantage is gained by dividing a problem into pieces, then the smaller the pieces the better, right? And if it gets too small, so that mutation on bits is sufficient, we can always mislead it → hence, fully-deceptive traps Exp in k is not a problem if k is small, and At least its not exponential in L, right? Well, yes – but this scenario doesn’t require recombination of good blocks… the smaller the blocks, the easier it is to do without BB recombination

8 blocks - 8 Richard Watson Crossover vs mutation without linkage info Uniform xover = ‘common good’ + macromutation parent A parent B ??????111111?????? offspring This utilises information from the population to ‘identify’ which parts to randomise – giving expected time exponential in k (not L), regardless of linkage. A mutation hill-climber couldn’t do this without linkage information. But it doesn’t transfer good blocks from one individual to another – there’s no building-block story required Overall time still exp in k (at best) → requires small blocks.

9 blocks - 9 Richard Watson Crossover vs ‘macromutation’ with linkage info E.g. two-point xover = ‘common good’ + recombination of segments that disagree parent A parent B AAAAA111111BBBBB offspring Utilises common good, AND if linkage is tight, expected time to find appropriate crossover points is better than L 2. Does transfer good blocks from one individual to another! However, if popsize exponential in k, then overall time is still exponential in k (at best) → requires small blocks. And, if use of linkage information is allowable, macromutation hill-climber is also order L 2 2 k i.e. pick two crossover points, randomise all bits in between.

10 blocks - 10 Richard Watson Reconsider assumptions Fully-deceptive trap function assumes optimisation at only the block scale is ideal. i.e. Selection on blocks is necessary and sufficient Selection on bits not sufficient and not necessary After all, to the extent that selection on bits is useful, selection on blocks seems redundant…? That is, if … selection on bits can find good blocks, and selection on blocks finds good genotypes then shouldn’t it be the case that selection only on bits is sufficient to find fit genotypes?

11 blocks - 11 Richard Watson … In order to show that selection on blocks is required Selection on bits must be insufficient. But maybe Selection on blocks may not be sufficient on its own either, and Selection on bits might be required too. Finding utility in selecting on blocks need not preclude some utility in selecting on bits. Can we utilise selection on bits to find good blocks, without precluding utility of selecting on blocks?

12 blocks - 12 Richard Watson Consider, partially- deceptive sub-function Hill-climbing (i.e. selection on bits) will reach one or the other optimum in time O(klogk) likewise, mutation and selection, even in a small population Prob. 0.5 of reaching high optimum, in each block. As before, two point recombination can bring good blocks together in time at worst L 2. (to find req. crossover points) There’s no need for any process that’s exponential in k, so we can use large k, without impeding the GA. k can be a constant fraction of L, and its all still polynomial (So long as good blocks are maintained in the population) Whereas hill-climber will take time exponential in L unitation 012… k fitness k-1

13 blocks - 13 Richard Watson Likelihood of hill-climber succeeding If HC doesn’t arrive at AB it will be doomed But if some indivs arrive at Ab and some arrive at aB, then they can be crossed to find AB But if you had enough diversity to find both Ab and aB you would have enough diversity to find AB! But this isnt true for many blocks the probability of finding B blocks all correct in one individual is exp. small in B whereas there’s a reasonable probability of having each block correct in at least one individual, even with small popsize Consider 2 blocks: There are 4 possible results of a local hill climber: ab, Ab, aB, AB.

14 blocks - 14 Richard Watson Given large k (constant fraction of L) Uniform xover fails, and macromutation (even with the use of linkage information) also fails, And more to the point, both of these become far removed from performance of two-point crossover with tight linkage, O(L 2 ) Large k is required to properly separate performance of algs that do not recombine building-blocks from performance of algs that do In concatenated trap functions with large k Selection on bits not sufficient but is necessary Selection on blocks is necessary but not sufficient Local optimisation at both scales is required

15 blocks - 15 Richard Watson So what? Partially deceptive trap functions with large k are (in principle) better at distinguishing ability of sexual population from asexual population and hill-climbers The contribution is more conceptual than technical – I suggest that thinking of a GA as a mechanism for manipulating blocks, has blinded to importance of also utilising selection on bits The root of the problem is that a block is not really a block (wrt selection) if there’s no selection ‘inside’ it –its just ‘atom’ with a big alphabet. And then there’s no point to it – fully-deceptive blocks  ‘block-wise one-max’ where each step is exp in k. But utilising two-scales of optimisation, local optimisation at bit scale (via mutational variation), and local optimisation at block scale (via recombination) can do something interesting mutation doesn’t merely provide variation for recombination to act on, it provides ability to follow selective gradients in nucleotide sequence space /= selective gradients in allele frequency space

16 blocks - 16 Richard Watson Generalising required properties of intra-block epistasis Doesn’t have to be as contrived as a ‘trap’ function in the sense of complementary optima Single-peaked wont do – selection on blocks is not required Random intra-gene landscapes won’t do – selection on bits is not useful for finding good blocks NKs won’t do either – direct trade-off between Utility of local adaptation vs Non-utility/insufficiency of local adaptation Necessary and sufficient: multiple peaks, with smooth-ish slopes, with significant separation between them

17 blocks - 17 Richard Watson Does that help with the evolutionary biology perspective of sex? Genes as blocks Usually think of blocks meaning groups of ‘genes’ But the blocks should contain many mutational units (not many genes) In natural genomes, mutational scale is very different from recombinational units

18 blocks - 18 Richard Watson Assumptions for an epistasis model of a natural genome Genes contain many nucleotides Nucleotides within a gene are strongly epistatic Epistasis between genes is relatively weak Thus, large disjoint sets of nucleotides are grouped both functionally and physically Intra-gene epistasis creates multiple local optima (in nucleotide-sequence-space) That are significantly distant from one another in nucleotide sequence space That have significantly different fitnesses Following selective gradients in nucleotide sequence space from a given ancestral sequence will not always result in discovery of the same local optimum

19 blocks - 19 Richard Watson Inter-gene interactions Assume that the fitness of a genotype, G, will be given by the fitness of each of the genes, g 1, g 2,…,g B, with no epistasis between genes (i.e. multiplicative fitness): where f(g i ) is the fitness contribution of the i th gene.

20 blocks - 20 Richard Watson e.g. Intra-gene landscape (i.e. one sub-function) The fitness contribution of each gene, g, will be an epistatic function of some nucleotides that it contains defined using P randomly positioned peaks: Where b p is a locally optimal sequence, ω p is the height of peak b p, and h(x) defines the shape of the peak

21 blocks - 21 Richard Watson Intra-gene landscape, 2D illustration Using 10 randomly positioned peaks with heights  j =1/(1+j) to create a range of heights.

22 blocks - 22 Richard Watson Intra-gene landscape; A couple of 1D illustrations Not as contrived as the trap sub-function – its just a multi- peaked sub-function Hill climbing from a random start position might not always reach the same peak.

23 blocks - 23 Richard Watson Overall landscape Represented using the product of two 1-D intra-gene landscapes

24 blocks - 24 Richard Watson Simulation parameters 25 genes of 25 bits each. 10 randomly positioned peaks in each gene. “if competing block solutions are maintained long enough”… i.e. first good-ish block doesn’t fix before others have chance to find a better one. population subdivision 30 demes of 10 individuals each. Rank selection Migration rate = 1 individual per deme per 10 generations is a migrant. (Converged initialisation) 30 runs per point; ave of ave, and ave of best

25 blocks - 25 Richard Watson Asexuals, to low-rate crossover, to uniform A low per-locus crossover rate, that can keep nucleotides of a gene together but assort genes, is preferred over asexuals (1000 fold) and over free recombination of nucleotides (10000 fold). Fitness after 1000 generations Crossover rate

26 blocks - 26 Richard Watson Block-wise crossover, and shuffled control Fitness after 1000 generations Linkage model

27 blocks - 27 Richard Watson Subdivision not essential

28 blocks - 28 Richard Watson Mutation sensitivity (block-wise crossover)

29 blocks - 29 Richard Watson observations Crossover enables large specific changes in genotype space Not small ones Not random ones Low mutation wont do High mutation wont do Both is OK in two phases Still needs linkage info (nearest basin boundary is k/2 bits away, if mutations affect many blocks, it’ll trade improvements for steps backward) Q what can a GA do that MMHC with two scales of mutation cannot? Do the things that are put together _need_ to be found in parallel? need Gap function!

30 blocks - 30 Richard Watson Conclusions Simple concat BB functions DO show adv of recombining blocks if blocks are large, not fully deceptive, and tightly linked Has ‘natural’ interpretation If you don’t need/cant use selection any selective gradients on parts within a block, then finding a block takes time exp in blocksize, → blocks must be small Although dividing a problem into small subproblems seems like a good idea, the other way to look at it is that the smaller the block the better able is mutation to find the block by chance (and the less essential is the combination of blocks)

31 blocks - 31 Richard Watson Conclusions 2 The benefit of sexual recombination in natural genomes may derive simply from their most basic genetic architecture: the fact that genomes contain thousands of functionally and physically particulate genes, each composed of thousands of strongly epistatic nucleotides.

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36 blocks - 36 Richard Watson Motives from Evolutionary Biology In population genetics, sex defines the units of selection. and forces selection to act on these ‘particulate’ units… specifically, genes (not organisms, not genotypes, hence ‘selfish gene’) Thus, evolution = movement in allele freq space In Evolutionary Biology view (both Wright and Fisher) Alleles of different genes are essentially unlinked (freely recombining=uniform xover) If they’re tightly linked then they behave as if a single allele There is never any utility in discussing selection on combinations of alleles – either a pair of alleles recombine or they’re one allele Is that view correct?

37 blocks - 37 Richard Watson Contrast with EC Some kinds of EC propose sex as a means to manipulate ‘building-blocks’. – to select on ‘composite’ things. In a building-block function, we suppose that Selection on bits not sufficient Selection on blocks is necessary Ideally(?), precluding the utility of selection on bits, as in fully-deceptive trap functions, should secure the requirement and maximise advantage of selecting on blocks But that just means that the blocks are really macro ‘atoms’ – like an allele with 2 K alleles

38 blocks - 38 Richard Watson No use for block concept? If selection acts only on blocks – then they’re not really blocks (wrt selection) If selection acts on bits, then it cant act on blocks as well Maybe there’s no use for block concept? Is there ever any circumstance where two-scales of selection are required to understand what’s going on? – forcing us to treat one of these scales as a combination of units from the lower scale, rather than an macro unit.

39 blocks - 39 Richard Watson Interaction of recombination and mutation Everybody knows that (even if you believe that recombination does something clever and important with blocks) mutation is required to provide the variation for recombination to act on. Is that all it does? Note that recombination and point mutation in biological systems operate on units at very different scales: Spontaneous point mutation facilitates movements in nucleotide sequence space Recombination manipulates alleles, each containing thousands of nucleotides; hence facilitates movement in allele sequence space What if mutation doesn’t merely provide variation for recombination to act on, it provides ability to follow selective gradients in nucleotide sequence space /= selective gradients in allele frequency space Interaction of mutation and recombination → two scales of optimisation

40 blocks - 40 Richard Watson Simple analysis: with recombination k block 1 block  --- block 1 block  block 1 block block 1 block 2 T k

41 blocks - 41 Richard Watson For pop gen Pop gen of sex view depends on alleles being mutational neighbours – here neighbours in crossover variation are a subset of mutational neighbours No neighbours in allele seq space that are not neighbours in nucleotide seq space But natural alleles differ at many nucleotide sites

42 blocks - 42 Richard Watson notes It means we can use a simplified linkage model: inter- block crossover points only (no partial linkage nec.)

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46 blocks - 46 Richard Watson So, can selection operate on both bits and blocks?

47 blocks - 47 Richard Watson Either genes are tightly linked and behave as if a single allele, or they’re unlinked Blocks are only meaningful to selection if they’re heritable – and bits are only meaningful to selection if they’re heritable (individually). If sex defines the unit of selection, does that mean the internals of building-blocks are irrelevant? In order for a composite to be a meaningful composite, its parts have to be significant. However, recombination and spontaneous point mutation operate at quite different scales. Is variation at these two different scales (together) the key to unifying EC BB view of sex with EB view that sex defines the units of selection?

48 blocks - 48 Richard Watson Are natural genes ‘atomic’ or ‘composite’ things? Are ‘genes’ in a GA ‘genes’ or ‘nucleotides’? If sex defines the unit of selection, are the internals of the gene irrelevant? Are building-blocks in a GA ‘genes’ or ‘groups of genes’? For a block to be group, it must have meaningful component parts.

49 blocks - 49 Richard Watson Overview (Supposed) Preconditions for utility of selecting on blocks Selection on bits not sufficient and not necessary Selection on blocks is sufficient and necessary An old favourite (that does not require BB combination!) Concatenated trap functions (fully-deceptive, with small k) Preserving common alleles + macro-mutation (exp in k) Uniform vs two-point Concatenated trap functions with large k Selection on bits not sufficient but is necessary Selection on blocks is not sufficient but is necessary Local optimisation at both scales is required A slightly less contrived (more believable) example

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55 blocks - 55 Richard Watson Overview2 (Supposed) Preconditions for utility of selecting on blocks Selection on bits is insufficient (and not required) Selection on blocks is sufficient An old favourite (that does not require BB combination!) Concatenated trap functions (fully-deceptive, with small k) Preserving common alleles + macro-mutation (exp in k) Uniform vs two-point Concatenated trap functions with large k Selection on bits is required (but insufficient) Selection on blocks is insufficient (but required) Local optimisation at two-scales A slightly less contrived (more believable) example

56 blocks - 56 Richard Watson Overview What do population genetics models for the benefit of sexual recombination look like Linkage disequilibrium Fisher/Muller Muller’s Ratchet Deterministic mutation hypothesis Contrast with computational ideas for benefit of sex Why are they so different? Selection on individual alleles vs combinations of alleles Wright vs Fisher A simple model where selection on more than one level of unit is required What this means for Wright vs Fisher and benefit of sex

57 blocks - 57 Richard Watson Recombination of EC and EB Evolutionary Computation (EC) Genetic Algorithms Evolutionary Algorithms Evolutionary Biology (EB) Population Genetics Evolutionary Genetics Recombination (Crossover) Evolutionary Computation Evolutionary Biology

58 blocks - 58 Richard Watson Population genetics The study of genetic variation within a population The mathematical, formal kind of evolutionary biology Origins in The Modern Synthesis, Fisher, Haldane, Wright 1930s Between Darwinian theory of evolution and Mendelian genetics Solved the problem of loss of variation under blending inheritance Research questions such as How does the frequency of a beneficial allele in a population change over time? What is the probability that it goes to ‘fixation’ or is ‘lost’ from the population? What would you expect the variation in a population to look like if it was all neutral?

59 blocks - 59 Richard Watson Sexual reproduction diploid parent cells meiosis parent one parent two haploid gametes syngamy diploid offspring cell

60 blocks - 60 Richard Watson Sexual recombination parent one parent two Chromosomal reassortment

61 blocks - 61 Richard Watson Sexual recombination crossover parent one parent two

62 blocks - 62 Richard Watson What’s the difference really? Both chromosomal reassortment and crossover result in sexual recombination – new combinations of genes in the haplotypes If genes are on different chromosomes they recombine with probability 0.5 If genes are on the same chromosome they recombine with probability between 0 and 0.5 depending on their ‘genetic distance’

63 blocks - 63 Richard Watson ‘Crossover’ in genetic algorithms parent one parent two parent one parent two One-point crossover Uniform crossover

64 blocks - 64 Richard Watson What is recombination good for? Well what does recombination do? It reduces the probability that two alleles in a parent both appear in the offspring… Or it creates the possibility that two alleles that did not both appear in either parent may appear together in the offspring… One is just the flip-side of the other…

65 blocks - 65 Richard Watson Genetic linkage, linkage (dis)equilibrium Linkage equilibrium: joint frequencies of alleles are product of marginal frequencies i.e. f(AB)=f(A)*f(B) i.e. no particular combinations of alleles are over or under represented Linkage disequilibrium (genetic linkage) Any deviation from the above Physical linkage The tendency of alleles to travel together during recombination due to physical proximity on the chromosome Under uniform crossover (free recombination) there is no physical linkage Returns alleles to linkage equilibrium The more recombination – the less linkage disequilibrium

66 blocks - 66 Richard Watson Population Genetics and sex Bear in mind that reversion to asexual reproduction (parthenogenesis) is readily available in most species (except mammals) Pop. Gen. models for the benefit of sex have many flavors e.g. Single locus models, rapid adaptation to changing or oscillating environment (esp. parasites), DNA repair. I’m going to focus only on 3 that involve adaptive benefit of recombination If sex reduces linkage disequilibrium: then for a benefit we need a) a reason for LD. b) a benefit to reducing it.

67 blocks - 67 Richard Watson Fisher/Muller hypothesis (1930) A B A B A+B Linkage disequilibrium created by new mutations in finite populations. Decrease in L.D. beneficial because removes competition between simultaneously segregating beneficial alleles i.e. sex is advantageous because it allows alleles to be selected for independently See also: ‘traffic problem’, clonal interference, Hill and Robertson effect. time Proportion of population

68 blocks - 68 Richard Watson Muller’s Ratchet In finite populations with mutation If every copy of a good genotype has acquired one or more deleterious mutations then an asexual population cannot recover the superior wild type. One click of the ratchet. But a sexual population can cross two half-unmutated individuals to recover an unmutated individual. Assumes no elitism, no back mutations, (no beneficial mutations). Complement of Fisher/Muller: combination of good non-mutants vs combination of good mutants (start in high frequency vs start in low frequency) Pro Deleterious mutations more common than beneficial mutations (co-occurrence more likely) Cons Both models only apply to finite pops Charlesworth: time for a click VERY long on large pops.

69 blocks - 69 Richard Watson Deterministic mutation hypothesis (Kondrashov 1989) Assume negative epistasis: two mutants worse than twice as bad as one L.D. created because selection removes double mutants faster, giving over-representation of single-mutants Reducing L.D. is good because: this L.D. decreases variance in mutation number, reducing effectiveness of subsequent selection (if you had double-mutants you could get rid of mutations quicker) Recombination decreases L.D., (recreating double mutants) increasing efficacy of selection Pros. Deleterious mutations more common than beneficial mutations (co- occurrence more likely) Applies in infinite pop size (no drift) hence ‘deterministic’ Empirical investigations underway to measure epistasis of single and double mutants

70 blocks - 70 Richard Watson From an EC point of view There’s no problem here from EC point of view. There are no local optima – hill-climber or population with elitism makes all issues disappear In EC we look for scenarios showing principled difference in time to find fittest genotypes (e.g. polynomial versus exponential) Something that a simpler algorithm e.g hill-climber couldn’t do

71 blocks - 71 Richard Watson A slightly more general view? Pop gen models view sex as reducing linkage disequilibrium Enabling selection to act on individual alleles Alternatively, we can see sex as defining the unit of selection – a la Dawkins, Williams In pop gen this is taken to define what an individual allele is, but in evolutionary computation we do not think of it as ‘atomic’ The building-block hypothesis says that GA works well when it does because it searches combinations of ‘schema’ Tightly physicially linked alleles… but not individual alleles

72 blocks - 72 Richard Watson What’s an allele anyway? Pop. Gen. models for benefit of recombination follow assumption of free recombination (uniform crossover) Obviously, there isn’t free recombination of nucleotides – physical linkage permits selection on sets of nucleotides In pop. gen. an allele is defined as the particulate unit of inheritance If loci are not physically linked then selection acts on individual loci If loci are physically linked then treat as one locus Either way, selection acts on of individual ‘alleles’

73 blocks - 73 Richard Watson The benefit of recombination in EC It’s been difficult to show any simple case where an EA with crossover outperforms and EA without crossover in a principled manner Holland, Goldberg, Mitchell and Forrest, Jones, Vose… I’ve done some work on this (Watson )

74 blocks - 74 Richard Watson Prior work For a sexual population Different individuals discover different modules Crossover exchanges whole modules between individuals shows fundamental distinction in EC: poly vs exp in size of system → large systems unevolvable to asexual popn. Example of general class of ‘compositional mechanisms’: sex, lateral gene transfer, symbiogenesis But these models are too complex/weird for Pop. Gen. audience Fitness landscape too esoteric Details of algorithm (crowding method) non- biological P1: P2: C1:

75 blocks - 75 Richard Watson What’s an allele anyway? In EC we have the idea of finding solutions to building blocks and then putting them together – an essentially two-level process But pop gen definition suggests that recombination rate defines the unit of selection – there cannot be two levels Why might we need two-levels? What are the bits in a genetic algorithm? Nucleotides? Alleles of genes? Does it matter? Consider mutation on nucleotide sites and recombination of genes… Suppose mutation on nucleotide sites finds good alleles for a gene and recombination of genes finds good genotypes. Is a two-level description of selection ever necessary? If selection on individual sites finds good alleles then more of that finds good genotypes, no?

76 blocks - 76 Richard Watson Two module system gene 1 gene 2    i j Nucleotide sites within a gene are grouped both functionally and physically by virtue of shared transcription and translation. → A simple form of biologically-real modularity. Assume mutations in the same gene have stronger synergy than mutations in different genes i.e. more beneficial mutations there are within a gene, the more effect additional mutations have. Equivalently, the closer to proper/complete functioning the gene is, the more important mutations are. e.g. each deleterious mutation halves binding affinity - exponential decay with distance from ideal sequence

77 blocks - 77 Richard Watson Multiplicative in gene 1 only e ie i

78 blocks - 78 Richard Watson Synergistic epistasis within genes (e i +e j )

79 blocks - 79 Richard Watson F(G) beneficial mutations in gene 2 ( j ) beneficial mutations in gene 1 ( i ) R i,j Random epistasis among all loci

80 blocks - 80 Richard Watson Synergistic epistasis within genes and random epistasis (e i +e j )R i,j

81 blocks - 81 Richard Watson 2k2k Simple analysis: without recombination k k gene 1 gene 2 Finding peak of one gene is easy. Finding the peak of the second gene without disrupting the first is difficult. T 2 k

82 blocks - 82 Richard Watson Simple analysis: with recombination k gene 1 gene  --- gene 1 gene  gene 1 gene gene 1 gene 2 T k

83 blocks - 83 Richard Watson Simulation study Sufficient diversity required. Multi deme model 50 demes of 200 individuals each Island migration (0.005) Mutation (0.03 per site) Recombination: crossover 0.0, (1/L), 0.5 Two genes, 40 sites each 1% Elitism: retain fittest 2 individuals in each deme Simulation example…

84 blocks - 84 Richard Watson Simulation results

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87 blocks - 87 Richard Watson In this two-module model We cannot abstract combinations of sites into alleles of genes Although they’re particulate under recombination, Because these alleles not particulate under mutation (mutation on sites is required to find fit alleles for genes, but not sufficient (by itself) to find fit genotypes) i.e. To see this effect we need both Selection on individual site mutations to find good alleles And selection on combinations of sites (alleles of genes) to find good genotypes i.e. find solutions to blocks and them put them together Selection on either one level is insufficient, cannot abstract combinations of sites into alleles

88 blocks - 88 Richard Watson Wright vs Fisher: on epistasis, sex, the unit of selection Wright: Epistasis creates local fitness peaks, central problem of evolution is escaping local optima Fisher: A) epistasis doesn’t create peaks in multi-dimensional landscapes B) even if it did, evolution has no way to select on combinations of alleles (because of sex) so selected change will always be a reaction to average excess of individual alleles (in the current population) and nothing more So: Epistasis is only variation in average excess of an allele, Evolution is change in freq of alleles in a population Wright: A) ? (Kauffman shows that expected number of peaks in random landscape is 2^L/L!) B) Shifting balance theory (SBT): in a subdivided population, different subpopulations (demes) may select for different combinations of alleles Big problems with SBT Fisher: I win! :) - average excess of individual alleles governs evolution

89 blocks - 89 Richard Watson Unit of selection (in population genetics) Both Fisher and Wright, (and Williams and Dawkins) follow the assumption that because sexual recombination breaks up combinations of alleles (and combinations of alleles are therefore not heritable) selection cannot act on combinations of alleles (in a panmictic population). Selection must act on individual alleles (genes not genotypes/individuals) → Mass selection: Evolution is change in freq of individual alleles, macro evolution is just this many times, at many loci. What about epistasis?: For sure changes in genetic background change fitness effects of an allele and thus alter its subsequent changes in freq, but what can you say about that? Evolution is cannot anticipate it, and thus neither will I. Not: there is no epistasis. But: epistatic interaction systems can’t be selected for so I’m going to treat epistasis as noise (not Wright).

90 blocks - 90 Richard Watson A personal view of the big picture Why are EC and Pop. Gen. so different? Darwinian assumption of gradual change, A theory of evolution by natural selection needed an optimisation algorithm that could be implemented in natural mechanisms Darwins model was a hill-climbing model Hill-climbing is the simplest algorithm Fisher reinforced/formalised a gradual change framework Although epistasis may be present, only need to model additive effects → models with small number of loci with simple epistasis (or none)

91 blocks - 91 Richard Watson … In EC we know that accumulation of small changes can only find good solutions if epistasis is simple – (to the extent that evolution behaves like a hill-climber its not interesting to EC) But that’s OK for biology – natural evolution isn’t expected to find best solutions – only to show some propensity for adaptation But – there are other algorithmic possibilities…

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94 blocks - 94 Richard Watson Shifting balance theory (Wright 1939, 1971) Wright suggested that evolution in a subdivided (multi- deme) population would differ from that of a panmictic (freely mixing) population. Three phases assume population is stuck on a peak… I)Each deme drifts around a bit - because drift (random variation in allele frequencies from sampling error) is greater in small populations II)Some deme finds fitter combination of alleles (another peak) and increases in abundance III)Increased migration from fitter/bigger deme recruits less fit demes …thus population as a whole escapes local peak and arrives at fitter peak.

95 blocks - 95 Richard Watson …Shifting Balance Theory Two important aspects to the theory: Each subpopulation will be subject to greater genetic drift Selection may act on the combination of alleles in a subpopulation (which Wright called ‘interaction systems’) There’s free recombination within each deme but not among demes So different demes have different genetic make-up, and the growth/decline of different demes thus enables selection on allele combinations Problems: Balance of selection pressures and population sizes – population must be small enough to drift off first peak but big enough to select for better peak.

96 blocks - 96 Richard Watson Population Genetics in contrast with EC Population geneticsEvolutionary Comp. Focus on small models and simple epistasis Large systems and complex epistasis Single locus two-allele models, or infinite alleles (single locus) models Poly locus two-allele Freq of existing allelesnew mutations, discovery of genotypes Microscopic allele-centric/ Selection on alleles macroscopic genotype-centric / Selection on genotypes Time scale of population size generations Long timescales

97 blocks - 97 Richard Watson Why might EC and Pop. Gen. be so different? Pop. Gen.EC Different motivesnature as it isproblem solving ToolsLack of numerical methods/ necessity of analytic results intrinsically computational methods Styles of science vs engineering rigorous (uptight)Creative (sloppy) Historical contingency: types of questions that were pertinent take field in different directions All of the above, and something deeper? Selection on individual alleles vs selection on genotypes (combinations of alleles)…

98 blocks - 98 Richard Watson Compare with EC Background assumption that we’re doing combinatorial optimisation and looking for fit genotypes (not so strong in ALife) Mutation hill-climbing as first approximation for evolution We know that an EA with crossover potentially changes this: Holland suggests selection on schemata Given free recombination (uniform crossover) → schemata are ‘order 1’ ≡ Pop. Gen. framework. (see PBIL - Baluja 1994 ) But one-point or two-point crossover → schemata are many alleles We assume that sex is allowing selection on combinations of alleles (building blocks), not preventing it! In fact, low rates of recombination in an EA prevent selection on whole genotypes but enable selection on small groups of alleles But the ‘alleles’ that Pop. Gen. is talking about are defined as particulate heritable units – schemata scale.

99 blocks - 99 Richard Watson Conclusions to part 3 Pop. Gen. focuses on change in frequency of individual alleles This is because natural selection cannot act on combinations of alleles that are broken-up by recombination SBT was the only serious attempt to find an exception – but it is generally not accepted In EC we focus more on search for fit genotypes (not changes in freq of alleles) We know that crossover potentially changes the unit of selection but we have a very different view of its impact …

100 blocks Richard Watson … Conclusions to part 3 So there are very different emphases Many reasons for this including: a desire to answer the burning question of blending inheritance and the maintenance of variation in a population (on very short timescales) vs combinatorial optimisation (more akin to evolution writ large) Accept our differences still valuable exchanges – e.g testing for neutrality of evolutionary change in ALife systems an understanding of the differences is a first step to building bridges But there are also actual disagreements here about how evolution works and what it can do Next section: look deeper into one of the critical issues at the heart of the different views… Sex and the unit of selection I think there’s a chance here to use EC thinking to challenge Pop. Gen. thinking and make valuable contribution from EC to Pop. Gen.

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105 blocks Richard Watson EB in contrast with EC Large systems with complex frustration of variables Small models with simple epistasis ECEB Time (prob.) of finding fittest genotypeTime (prob.) to fix an allele (given that it arises/is present) Evolution as an optimisation method: How to find good (best) candidate solutions from the space of all possible candidate solutions Evolution as a dynamic process altering frequencies of alleles in a population: How genetic make-up of population changes over time Genetic Algorithm, GA: (Holland 1975) following Wright/Fisher model Population genetics: Fisher/Haldane/Wright 1930s

106 blocks Richard Watson Why such different emphases? Engineering motives vs Science: solve hard problems vs natural evolution is just what it is ? P.G. pre-computers → analytic statistics, difficult to apply to large systems vs. E.C. applications easy to apply to large systems ? E.B.: Darwin assumption of gradual change, Fisher formalised selection for average excess of an allele → evolution ≈ hill-climbing E.C.: hard problems cant be solved by hill-climbers, if evolution is hill-climbing what’s the point? Recombination → hill-climbing (E.B.) → not hill-climbing! (E.C.)

107 blocks Richard Watson Recombination in E.C. What can a Genetic Algorithm, G.A., do that a hill-climber cannot do? Population → population of hill-climbers, random re-start HC? Recombination → exchange of co-adapted subsets of variables Holland: Schema Theorem (1975) Increase in frequency of schemata of above average fitness Goldberg: Building Block Hypothesis (1989) Emphasis on combining fit low-order schemata into higher-order schemata of higher fitness Mitchell and Forrest, Jones ( ) – Royal Roads and Headless Chicken test → general failure to demonstrate anything a GA can do that a non- population-based mutation/selection algorithm cannot do.

108 blocks Richard Watson Recombination in E.C. Hierarchical-IFF: Watson (2000) Hierarchical modular building-block problem Hill-climber O(e N/2 ) GA with tight linkage can solve in O(N 2 log 2 N) Recursive divide and conquer problem decomposition P1: P2: C1: P1: P2: C1: Jansen (2001) ‘Gap’ function (non-recursive) Limitations Both example landscapes very complex/contrived for the purpose of illustrating the distinction Both use modified GAs : diversity maintenance Neither has direct biological correlate

109 blocks Richard Watson Recombination in E.B.: classical model Fisher/Muller (1930) Formalised by Hill and Robertson 1969 A B A B A+B Requires: interval between ben mutations to be small wrt time to fix a beneficial mutation - small selection coeficients - large populations For EC: a HC is best (beneficial mutations can be accumulated sequentially time Proportion of population

110 blocks Richard Watson Recombination in EB: deterministic mutation hypothesis ‘deterministic mutation hypothesis’ (Kondrashov 1989) (‘deterministic’ - Infinite population, all alleles present at all times) Focussing on purging of deleterious alleles ‘Synergistic Epistasis’– two deleterious mutations worse than twice as bad as one deleterious mutation Creates linkage disequilibrium: single-del-mutants over-represented Recombination reduces linkage disequilibrium: double-del-mutant frequency increased Selection against double-del-mutants more than double the selection against single- del-mutants Therefore deleterious alleles are purged from the population more rapidly in sexual population Pros Deleterious mutations more common than ben. mutations (co-occur. more likely) Empirical investigations underway to measure epistasis of single and double mutants EC: fittest genotype already present in population (and if not HC can accumulate beneficial mutations sequentially)

111 blocks Richard Watson Summary of prior work EB Two-locus two-allele exemplars Even with more loci: All examples are single-peaked so HC can solve in time linearly proportional to number of mutations required No example where recombination allows evolution of something that’s unevolvable with asexual population. Only: W/F: faster (prop. to size of pop.) to reach double mutant DMH: Increase freq. of beneficial alleles at equilibrium EC Has shown exponential vs polynomial (in size of problem) Contrived/complex No direct biological correlates

112 blocks Richard Watson Unidimensional and multidimensional epistasis All EB models so far (Mutations are unambiguously del. or ben.) Unidimensional epistasis models Counting number of mutations in the genotype Multidimensional epistasis (m.d.e.) models Cant be modelled by counting mutations

113 blocks Richard Watson Recombination in E.B. – multidimensional epistasis Multidimensional epistasis and the disadvantage of sex (Kondrashov and Kondrashov 2001): count mutations in two disjoint subsets of sites The fitness ridge used in K&K Number of 1-alleles at loci from the 1st set Number of 1-alleles at loci from the 2nd set Time to reach point (18,18) is twice as long for sexual population and asexual population 1-allele in second set may arise in beneficial background (i.e. exactly 6 1-alleles in first set) and this genotype may rise in freq in asexual popn. But this allele may be deleterious in background of other individuals in pop. and not rise in freq in sexual populations until popn. fixed for exactly 6 1- alleles. F(G)=1.1 n 0.7 D 1 st set 2 nd set    i j

114 blocks Richard Watson Varying crossover rate crossover rate (log scale) generations (log scale) Kondrashov m.d.e. – sensitivity Varying mutation rate & population size

115 blocks Richard Watson crossover rate (log scale) generations (log scale) K&K model Multidimensional epistasis – variations of K&K model Sensitivity to placement of gaps With fork With gaps The fitness ridge used in K&K Number of 1-alleles at loci from the 1st set Number of 1-alleles at loci from the 2nd set Fork and large gap P1: P2: C1: ?00??0000?00?0?000 0?0000??0000??00?0

116 blocks Richard Watson A simpler multidimensional epistasis model For EB and EC

117 blocks Richard Watson No modularity – all sites behave the same

118 blocks Richard Watson dependencies in gene 1 only

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120 blocks Richard Watson Resultant modularity

121 blocks Richard Watson Modular with random noise (e i +e j )I i,j

122 blocks Richard Watson 2k2k Simple analysis: without recombination k k gene 1 gene 2 Finding peak of one gene is easy. Finding the peak of the second gene without disrupting the first is difficult. T 2 k

123 blocks Richard Watson Simple analysis: with recombination k gene 1 gene  --- gene 1 gene  gene 1 gene gene 1 gene 2 T k

124 blocks Richard Watson Simulation study Sufficient diversity required. Multi deme model 50 demes of 200 individuals each Island migration (0.005) Mutation (0.03 per site) Recombination: crossover 0.0, (1/L), 0.5 Two genes, 40 sites each 1% Elitism: retain fittest 2 individuals in each deme Simulation example…

125 blocks Richard Watson Simulation results

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128 blocks Richard Watson Biological plausibility epistasis sites Assumptions. Sites divide into subsets (genes) such that: - Epistasis is stronger intra-gene than inter-gene. - Physical linkage stronger intra-gene than inter-gene. gene 1 gene 2    i j Possible to show effect on landscape w/o noise With converged initialisation Without elitism

129 blocks Richard Watson Why ignore genetic map at molecular scale? Molecular genetics history Assumption that If loci are not physically linked then selection acts on loci If loci are linked then treat as one locus Either way, selection acts on (average excess) of individual ‘alleles’ However, although alleles of genes are particulate under recombination they are not under mutation Combination of mutation on sites and recombination of genes together cannot be accommodated in single level model Certain kinds of epistasis make fit genotypes unevolvable for asexual populations and yet easily evolvable for sexual pops.

130 blocks Richard Watson Fisher vs Wright Both Fisher and Wright believed that fitness landscapes contained epistasis But, Fisher argued that local optima were rare in high-dimensional spaces (which is wrong) that even if there were local optima, selection cannot act on combinations of alleles (because recombination disrupts them) Wright had to show a mechanism where selection on combinations of alleles was possible Shifting Balance Theory: selection in subdivided populations not the same as selection in panmictic population But, relies on particular selection coefficients and population sizes Here, No need for genetic drift (works with deterministic selection within demes) Selection on parts and wholes via variation from mut. and rec. (not via selection within demes and among demes) Particular to correspondence of genetic map and epistasis

131 blocks Richard Watson Message for EC Recombinative population ≠ hill climber(s) Divide and conquer problem decomposition Find two parts and then combine Exp. vs poly. time Simple demonstration of building block recombination Not hard to imagine this form of modularity in engineering problems Two different subsystems are easy to find individually but having found one, variation that improves the other without disrupting the first is rare Note that a problem can be modular and still have important dependencies between modules: the dependencies between modules prevent good modules from accumulating sequentially (otherwise HC is superior) Potential to exploit modularity in problem domains However Result dependent on ‘tight-linkage’ Previous work on HIFF has shown that discovery of linkage map is possible Not yet attempted in this problem class

132 blocks Richard Watson Conclusion: Recombination in EC and EB Why is it the mechanism of recombination that provides a good domain to exchange results between EC and EB? In EC: looking for ways that GA ≠ hill-climber Recombination: divide-and-conquer problem decomposition In EB: deeply rooted assumption that evolution is a hill- climbing process Recombination ensures that selection acts on individual alleles → over-simplified models of epistasis What was needed was a simple demonstration of how evolution ≠ hill-climbing Variation (and selection) at more than one scale

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138 blocks Richard Watson Synthesis of EC and EB Variation at more than one scale missed by early PG Not just that neighbourhood space is different, or varies with popn – but that it is a dimensional reduction of the search space In HIFF its from bits to blocks with two solns each Here its from a large set of alleles to a smaller set of alleles Why should good alleles be selected for but yet we are unable, without recombination, to discover good genotypes? Because you need to search combinations of them Or because there is intereference between them – diminishing accessibility

139 blocks Richard Watson Preferred results Time to peak for various C Using max eval limit Even if its only 3 runs each or something I should do c=0.016 first to set eval limit low. Using proper island model w. low migration Using with and without noise ?time to fix peak? For k=30 only Kondrashov code doesn’t have subdivision My code outputs 30 runs for one C, does use (an almost standard form of) subdivision – fine.

140 blocks Richard Watson Differences for Bham vs Oxford For oxford Skip EC background, go straight into m.d.e Interpretation as speciation/hybridisation New m.d.e results without noise Why is evolution constrained on single peak? Why has it been missed before: linkage map, nonuniform epistasis For Bham Background on R in EB and EC: HIFF and Gap func Mde results with noise Emphasise impact for EC Show Time vs k Limitations: linkage map

141 blocks Richard Watson Classical Fisher/Muller story A A+B A B Second beneficial mutation appears (in parallel) in different background and is brought together by recombination. Second beneficial mutation appears (serially) in context of first. The former is likely only if simultaneously segregating beneficial alleles are likely – i.e. Mutations fix slowly wrt time for mutations to arise. (Fix slowly → Small selection coefs., large popn.). Classically, combinations that could appear by recombination, could appear by mutation alone.

142 blocks Richard Watson Classical Fisher/Muller story A B A B A+B A B

143 blocks Richard Watson With noise (gene 1)

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147 blocks Richard Watson Simulation results

148 blocks Richard Watson Varying crossover rate crossover rate (log scale) generations (log scale) K&K model Kondrashov m.d.e. – sensitivity Varying mutation rate & population size Placement of gaps With gaps

149 blocks Richard Watson Recombination in population genetics Fisher/Muller

150 blocks Richard Watson Conclusions Two-locus two-allele models cannot account for effects coming from large scale structure in the fitness landscape. Simple modular structure can be supposed from simple observations of genomics. Such modularity can have a profound effect on the benefit of recombination. This effect of recombination is not explained by existing two-locus models. The magnitude of the effect shown here grows exponentially with number of sites involved.

151 Acknowledgements Dan Weinreich - Harvard Alex Platt - Harvard Jordan Pollack - Brandeis

152 blocks Richard Watson Some inter-gene dependencies (gene 1)

153 blocks Richard Watson Some inter-gene dependencies (gene 2)

154 blocks Richard Watson Simple analysis: with recombination k gene 1 gene  --- gene 1 gene  gene 1 gene gene 1 gene  ----  --- T k

155 blocks Richard Watson Classical Fisher/Muller story A A+B A B Second beneficial mutation appears (in parallel) in different background and is brought together by recombination. Second beneficial mutation appears (serially) in context of first. The former is likely only if simultaneously segregating beneficial alleles are likely – i.e. Mutations fix slowly wrt time for mutations to arise. (Fix slowly → Small selection coefs., large popn.). Classically, combinations that could appear by recombination, could appear by mutation alone.

156 blocks Richard Watson Consider 4 loci Four genes with two alleles each: a/A, b/B, c/C, d/D – in that order on the chromosome ancestral genotype = abcd f(ABCD) > f(ABcd) = f(abCD) > f(Abcd) = f(aBcd) = f(abCd) = f(abcD) > f(abcd) > f([..others..]). Two ‘modules’ – AB and CD. AB and CD can be reached by one-point mutation But ABCD is mutationally isolated. But, ABcd can be crossed with abCD to reach ABCD without passing through a lower fitness state. cdCd/cDCD ab11+s1+2s Ab/aB1+s1-s AB1+2s1-s1+3s

157 blocks Richard Watson Fitness = K ℮ 1 ℮ 2 = C 1 C 2 = f 1 (g1) f 2 (g2) Fitness = K ℮ 1 ℮ 2 = C 1 C 2 = f 1 (g1) f 2 (g2) = f 1 (n 1,…,n 10 ) f 2 (n 11,…,n 20 ) Interpretation M1M1 M2M2 M7M7 M6M6 … g1 P1 C1C1 g2 P2 C2C2 I 21 I 12 I 21 I 12 ℮1℮1 ℮2℮2 K ℮ 1 ℮ 2 n1n1 n 10 n 11 n 20 Fitness = K ℮ 1 ℮ 2 = C 1 C 2 I 12 I 21 = f 1 (g1) f 2 (g2) f 3 (g1,g2) f 4 (g1,g2) = f 1 (n 1,…,n 10 ) f 2 (n 11,…,n 20 ) f 3 (n 1,…,n 20 ) f 4 (n 1,…,n 20 )

158 blocks Richard Watson Random: f 3 only

159 blocks Richard Watson Separable: f 1 and f 2 only

160 blocks Richard Watson Modular: f 1, f 2 and f 3

161 blocks Richard Watson Claims for discussion 1.The conception of modularity as nearly-independent sub-systems is often interpreted as though inter-module connections are unimportant: This is grossly misleading, and precludes multi-level hierarchical organisation. 2.Although modularity and hierarchical organisation are ubiquitous in the natural world/ biological systems, strong interdependencies between modules are also ubiquitous 3.Therefore, their evolution is not explained by accretive evolution 4.However, compositional evolution is fundamentally different, and can explain the evolution of systems that cannot be evolved accretively. Specifically hierarchically modular systems with significant inter- module dependencies can be evolved only by compositional evolution.


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